# Continuity and its relationship with asymptotic spectral decay

The asymptotic decay of the magnitude of the Fourier transform of a function appears always to be determined by its continuity properties as follows, with examples given in Fig. 1:

• Continuous integral, -20 dB / decade;
• Continuous function, -40 dB / decade;
• Continuous 1st derivative; -60 dB / decade;
• ...

How can this be expressed mathematically in a rigorous way, and proved?

Figure 1. dB magnitude of the Fourier transform of a function as function of the base-10 logarithm of frequency for (from slowest to fastest decay) a rectangular pulse (turquoise), a single lobe of a cosine (purple), and Hann function (blue).

I suppose it would be sufficient to show that a discontinuous function in the time domain decays as $$1/\omega$$ in the frequency domain (e.g., $$\textrm{rect}\Leftrightarrow\textrm{sinc}$$), and then use the fact that each integration in the time domain adds a multiplicative factor of $$1/j\omega$$ in the frequency domain.